On a visit to the Clark Art Institute in Williamstown, Mass., Benjamin Wiessner found himself admiring the museum’s prized early American silverware collection. Its shininess particularly captivated him. But then he fell asleep.

Of course, most visitors to the Clark are older than Mr. Wiessner, who was only 4 months at the time. His mother, Kimberly Ciola, had taken him to one of the museum’s talks for new parents and their babies.